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Question No. 4
There are two Vernier calipers both of which have 1 cm divided into 10 equal divisions on the main scale. The Vernier scale of one of the calipers (C₁) has 10 equal divisions that correspond to 9 main scale divisions. The Vernier scale of the other caliper (C₂) has 10 equal divisions that correspond to 11 main scale divisions. The readings of the two calipers are shown in the figure. The measured values (in cm) by calipers C₁ and C₂ respectively, are

(A) 2.87 and 2.86
(B) 2.87 and 2.87
(C) 2.87 and 2.83
(D) 2.85 and 2.82

Solution:

For both calipers:

  • Main Scale Division (MSD) = 1 cm / 10 = 0.1 cm

Caliper C₁:

  • Vernier scale has 10 divisions that correspond to 9 MSD.

  • So, Vernier Constant (Least Count) = (1 MSD) - (1 VSD)

    • 10 VSD = 9 MSD ⇒ 1 VSD = 0.9 MSD = 0.9 × 0.1 cm = 0.09 cm

    • Least Count (LC) = 1 MSD - 1 VSD = 0.1 cm - 0.09 cm = 0.01 cm

Caliper C₂:

  • Vernier scale has 10 divisions that correspond to 11 MSD.

  • So, Vernier Constant (Least Count) = (1 VSD) - (1 MSD) [since Vernier scale is longer]

    • 10 VSD = 11 MSD ⇒ 1 VSD = 1.1 MSD = 1.1 × 0.1 cm = 0.11 cm

    • Least Count (LC) = 1 VSD - 1 MSD = 0.11 cm - 0.1 cm = 0.01 cm

Both calipers have the same least count of 0.01 cm.

Reading the measurements from the figure (as described):

  • The main scale reading for both is between 2.8 cm and 2.9 cm. Specifically, the zero of the Vernier scale is just after 2.8 cm.

  • For C₁: The Vernier division that aligns with a main scale division is the 7th division.

    • Reading = Main scale reading + (Vernier division × LC) = 2.8 + (7 × 0.01) = 2.87 cm

  • For C₂: The Vernier division that aligns is the 7th division (same as C₁).

    • Reading = 2.8 + (7 × 0.01) = 2.87 cm

Therefore, both calipers give the same reading: 2.87 cm.

Final Answer:

2.87 and 2.87

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